Mr. T. S. Davies on Geometry and Geometers. 207 



and chronology. A few of his works (and few only) display 

 a certain amount of rough invention : but he was singularly 

 confused in his development of a process, and one of the most 

 uncouth of all the mathematical writers of this country. It is 

 very probable that, abating the Method of Increments (which 

 owed its value to its being then the only one in our language, 

 Dr. Brook Taylor's being in Latin and untranslated), his 

 most useful works have been the Elements of Geometry and 

 the Conic Sections. The former contains a few theorems 

 which are, as far as I know, original*; and the latter is va- 



* It is often extremely difficult to decide respecting originality in ele- 

 mentary investigations ; for no one can undertake to look carefully through 

 every elementary book that has been published, to ascertain whether some 

 particular and simple proposition might not possibly be contained in it. 

 Nevertheless some general criteria might be laid down, which would contri- 

 bute towards probability on one side or another in most cases ; and this 

 presumed probability would often limit the trouble of the search to very 

 narrow bounds. Almost every proposition naturally refers itself to a class j 

 and if once observed, others of that class must soon follow. If, then, upon 

 our observing such a proposition we find it isolated, it is highly probable 

 that it originated with the author who there gave it, or at least not long 

 before. In the few cases which I have had occasion to examine minutely 

 and carefully, I have rarely found this rule to fail — indeed, in no one to 

 signally fail. This, too, is precisely the same in respect to analytical de- 

 vices — and not widely different is the testimony of the history of experi- 

 mental science. 



This remark is made in consequence of a property of the triangle, now 

 universally known, which appears to have been first given in an elementary 

 treatise by Emerson (Geom., b. ii. pr, 32, 1763). "The perpendiculars 

 from the angular points of a triangle to the opposite sides, pass through 

 the same point." The property was, however, enunciated by Mr. Thomas 

 Moss, an exciseman and able geometer, in 1751 ; and two neat demonstra- 

 tions given to it shortly after by a writer who signs 20<l>02 (probably 

 Simpson), and by Edward Rollinson, in Turner's Mathematical Exercises. 

 A property still more general had been given four or five years previously 

 in the Mathematician, edited by Rollinson. The less general property 

 was not, however, perceived to be deducible from the more general one, 

 and both passed without further remark than the mere solutions till atten- 

 tion was called to the system of connected inquiries in the Mathematical 

 Repository (vol. vi.), under one aspect; and under another in the Phil. 

 Mag., vol. ii. p. 26,2nd Ser., and the appendix to the Ladies' Diary, 1835. 



It may seem strange that so simple a property, and so many others similar 

 or related to it, should have been unobserved by the antients, and by their 

 earlier followers after the revival of letters. It must be recollected, how- 

 ever, that the Greek geometers only valued a theorem (or even a Porism) 

 except so far as it contributed to the solution of a problem. There are 

 no traces, nor even intimations, of their having regularly attempted to form 

 classed collections of theorems, or to arrange systematically the many 

 beautiful series of properties of figures that could not fail to have presented 

 themselves during the solution of problems. Such properties were only 

 selected as would be actually required in demonstrating the constructions 

 of the cases of a problem. Of these the seventh book of Pappus is a col- 

 lection of instances. The properties of the apfirjkov given by him, form 



